Decibels

A decibel (abbreviated dB) is defined as one tenth of a bel. The bel^F.1 is an amplitude unit defined for sound as the log (base 10) of the intensity relative to some reference intensity,^F.2 i.e.,

$$\mbox{Amplitude\_in\_bels} = \log_{10}\left(\frac{\mbox{Signal\_Intensity}}{\mbox{Reference\_Intensity}}\right)$$

The choice of reference intensity (or power) defines the particular choice of dB scale. Signal intensity, power, and energy are always proportional to the square of the signal amplitude. Thus, we can always translate these energy-related measures into squared amplitude:

$$\mbox{Amplitude\_in\_bels} = \log_{10}\left(\frac{\mbox{Amplitude}^2}{\mbox{Amplitude}_{\mbox{\small ref}}^2}\right) = 2\log_{10}\left(\frac{\left\vert\mbox{Amplitude}\right\vert}{\left\vert\mbox{Amplitude}_{\mbox{\small ref}}\right\vert}\right)$$

Since there are 10 decibels to a bel, we also have

$$\begin{eqnarray*} \mbox{Amplitude}_{\mbox{\small dB}} &=& 20\log_{10}\left(\frac{\left\vert\mbox{Amplitude}\right\vert}{\left\vert\mbox{Amplitude}_{\mbox{\small ref}}\right\vert}\right) = 10\log_{10}\left(\frac{\mbox{Intensity}}{\mbox{Intensity}_{\mbox{\small ref}}}\right)\\ &=& 10\log_{10}\left(\frac{\mbox{Power}}{\mbox{Power}_{\mbox{\small ref}}}\right) = 10\log_{10}\left(\frac{\mbox{Energy}}{\mbox{Energy}_{\mbox{\small ref}}}\right) \end{eqnarray*}$$

A just-noticeable difference (JND) in amplitude level is on the order of a quarter dB. In the early days of telephony, one dB was considered a reasonable ``smallest step'' in amplitude, but in reality, a series of half-dB amplitude steps does not sound very smooth, while quarter-dB steps do sound pretty smooth. A typical professional audio filter-design specification for ``ripple in the passband'' is 0.1 dB.